2 answers

Let me denote 2*(x^3-2) by f and sign(0.5*x+2.0)*abs(0.5*x+2.0)^(1/3) by g.

Since f and g are inverse to each other (on the whole real line), there is a symmetry of the whole graphs with respect to the line {x=y}, but you do not see it.

First, you need to have the same scale for x and y axes, you can get this by setting the option aspect_ratio=1.

Second, you look at the interval [-4,0] for both functions while this interval is not invariant: f([-4,0]) is not equal to [-4,0], so the graph of g on [-4,0] has no reason to be symmetric of the graph of f on [-4,0]. If you want to see the symmetry, you need to plot the graph of g on the image of [-4,0] by f.